Remark

Here “SS” is supposed to be suggestive of a category of certain “geometric Shapes”. The canonical example is S=ΔS = \Delta, the simplex category, and the reader may find it helpful to keep that example in mind.

By restricting this to simplicial sets which are themselves simplicial nerves of categories (see below) or more generally are quasi-categories, this also induces the notion of geometric realization of categorical structures.

In fact, in that very article apparently what is now called Kan extension is first discussed.

Also, in that article, as an example of the general mechanism, also the Dold–Kan correspondence was found and discussed, independently of the work by Dold and Puppe shortly before, who used a much less general-nonsense approach.

In an article in 1984, Dwyer and Kan look at this ‘nerve-realization’ context from a different viewpoint, using the term ‘singular functor’ where the above has used ‘nerve’. Their motivation example is that in which SS is the orbit category of a group GG, and the realisation starts with a functor on that category with values in spaces and returns a GG-space: